We propose new antichain-based algorithms forchecking universality and inclusion of nondeterministic tree automata (NTA). We haveimplemented these algorithms in a prototype tool and our experimentsshow that they provide a significant improvement over the traditionaldeterminisation-based approaches. We use ourantichain-based inclusion checking algorithm to build an abstract regular treemodel checking framework based entirely on NTA. Weshow the significantly improved efficiency of this framework through a series of experiments with verifying various programs over dynamic linked tree-shaped data structures.